Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 13 x^{2} - 55 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0837222189534$, $\pm0.567942489704$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.18605.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5$ |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $75$ | $14625$ | $1649925$ | $210322125$ | $26018130000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $123$ | $1237$ | $14363$ | $161552$ | $1772343$ | $19480727$ | $214377763$ | $2358108337$ | $25937533398$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=8 x^6+8 x^5+7 x^3+8 x+10$
- $y^2=10 x^6+10 x^5+3 x^4+8 x^3+7 x^2+5 x+3$
- $y^2=9 x^6+7 x^5+3 x^4+8 x^3+4 x^2+2 x+10$
- $y^2=2 x^6+9 x^5+7 x^4+x^2+7$
- $y^2=8 x^6+x^5+4 x^4+7 x^3+4 x^2+6 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is 4.0.18605.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.11.f_n | $2$ | 2.121.b_afj |